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ANALYTIC GEOMETRY
ments; then $\frac{a}{d}=\frac{l_{1}}{l}$, $\lambda_{1}=\frac{l_{1}}{l}\lambda$, and the equations of the systems $P+\lambda Q=0$, $P_{1}+\lambda_{1}Q_{1}=0$ become $P+\lambda Q=0$, $lP_{1}+\lambda l_{1}Q_{1}=0$.
Now the elements of
$$ l\boldsymbol{P}_{1}+\lambda l\boldsymbol{Q}_{1}=\boldsymbol{0}... | c40aefa066f23dd013734396be5e162bba17af6ad084910ec23192a126bdf515 | us_public_domain_by_date | published 1897 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_dd50797fe0fbe3d6 | 25 | 15 | 7 | 3 | 46 | {
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隰所求之球中心为P，自P作已知二球之切線
PD, PE, 連結
AD, BE, 则球 P
直角截球 A, B,
因而 $ \overrightarrow{ADP} = \overrightarrow{BEP} = \overrightarrow{R} $，故 D, E，为球面 P
截二球面A，P，截口上之点，即PD，
PE，为球P之半径。今自P作垂直於
中心線之平面與AB之交點為C，則
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We have $ y^{2}= $DM. ME (Geom., Bk. IV., Prob. 23, Cor.); but by similar triangles we have
AB: AC:: AM: MD, whence MD= $ \frac{rx}{a} $;
also
AB: AC::BM: ME, whence $ ME = \frac{r}{a}(2a - x) $.
$$ y^{2}=\frac{r^{2}}{a^{2}}(2ax-x^{2}), $$
Whence
which is the equation of an... | 3cfe5f22a72d1662817a1b10c468ab2236789df77b1c16ecf610ebe36d2b3ac4 | us_public_domain_by_date | published 1900 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_73e59d114f83ac97 | 85 | 30 | 38 | 17 | 128 | {
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CONSTRUCTIONS DES FORMES PROJECTIVES.
Soient (fig. 47) A et A', B et B', C et C' les trois couples donné de Soient (fig. 48) a et a', b et b', c et c' les trois couples données de
Fig. 47.
Fig. 48. | 7ec48d4802863bf7289b798060ab7d2caf85bd61789cbe874eb30e9c894a8b5e | us_public_domain_by_date | published 1875 (US public domain by date); Wikimedia Commons; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_58a54c29a6cc0fa4 | 103 | 38 | 45 | 20 | 160 | {
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55
axis of the cycloid. The coordinates of any point of the cycloid may be expressed as transcendental functions of a variable angle.
Take the base of the cycloid as X-axis, the perpendicular to the base where the cycloid meets the base as Y-axis, and call the angle made by the radi... | ca7f42966eaebbe3bdbe8de41d02ebcc7784e747b08c934d3092e082bed4336f | us_public_domain_by_date | published 1897 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_1ab0299bcea82dd6 | 61 | 28 | 24 | 9 | 101 | {
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CHAPTER I
CO-ORDINATES. THE POINT
1. Application of Algebra to Geometry.—Let the student describe a circle of radius, say 5 inches, on any part of the board, and beneath it write the equation $ x^{2} + y^{2} = 25 $. We shall now ask the beginner whether he can perceive any possible relation between... | 196d0bf87564a1d4d39948bfcf50aa71630a876d9f48498ebd2b43cf06f26e34 | us_public_domain_by_date | published 1921 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_5d746f1f1405a60c | 45 | 17 | 21 | 7 | 67 | {
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INTRODUCTION
3
finally reaches, is the same as if, starting from OA, it had rotated through the single angle AOC. The angle
Fig. 3.
$ AOC $ is called the sum of the angles $ AOB $ and $ BOC $; that is, $ \angle AOB + \angle BOC = \angle AOC $. Evidently $ \angle AOB + \angle BOA = 0 $, and hence $ \ang... | d89058d20f17cd0b387576d5e5f1b5646ca2279143e61f8aa6baf3f1d4d99cce | us_public_domain_by_date | published 1901 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_1f79f39db5b9e70e | 26 | 10 | 12 | 4 | 37 | {
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hmb_00d75ff166cf8d25 | test | true | sg_c3804aedff | bk_555b85e5ee0c | wk_halsted_elements_of_geometry | Halsted - The Elements of Geometry | 1,885 | und | en | book | Archival (author-held scan) | not recorded (owner waiver) | 389 | null | repository plate image | [
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hmb_00e5b42442648428 | test | true | sg_c3804aedff | bk_75b06a64f496 | wk_shuxue_cidian | SSID-11317445 數學辭典 | 1,933 | und | zh | book | Wikimedia Commons | https://commons.wikimedia.org/wiki/File:SSID-11317445_%E6%95%B8%E5%AD%B8%E8%BE%AD%E5%85%B8.pdf | 484 | null | page-scan crop | [
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之使交弧AD於N，則N
二等分弧AD，故 $ \overrightarrow{DCM}= $
DBE, 故 $ \overrightarrow{CDM} = BDE $，故
$ \overrightarrow{ADE} = \overrightarrow{BDC} $
正方形 ABCD 之对角线，交点为
O，對角線AC上任意
一點爲P，過P平行於
邊之直線爲EF, GH,
則EB=CG, OB=OC,
$ \overrightarrow{OBE}=\overrightarrow{OCG}\left(-\frac{1}{2}\overrightarrow{R}\right) $，故 $ \Delt... | af9f9684001aa648443ebc4b59e2209a050b7551b7d283e6139cae76666c7489 | owner_determination | published 1933; US public-domain status recorded by the project owner; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_544ca6d4c369c92e | 89 | 28 | 51 | 10 | 113 | {
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hmb_011f8308f85c8399 | test | true | sg_2770a91e4b | bk_0de514b2f707 | wk_recorde_pathway_to_knowledge | Recorde - The Pathway to Knowledge | 1,551 | und | en | book | Project Gutenberg | https://www.gutenberg.org/ebooks/33093 | 168 | null | transcriber-prepared image | [
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hmb_0142b00d954dcc84 | test | true | sg_c3804aedff | bk_555b85e5ee0c | wk_halsted_elements_of_geometry | Halsted - The Elements of Geometry | 1,885 | und | en | book | Archival (author-held scan) | not recorded (owner waiver) | 470 | null | repository plate image | [
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hmb_017dd966b354b371 | test | true | sg_223423ea3c | bk_32a0331285eb | wk_nattes_s_practical_geometry_or_introduction_to_perspective | Nattes's practical geometry, or, Introduction to perspective | 1,805 | und | en | book | Internet Archive | https://archive.org/details/nattesspractical00leclrich | 107 | null | page-scan crop | [
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hmb_01918df0ebd3bb06 | test | true | sg_deb2de63f6 | bk_3fed6fb87a89 | wk_nam_1890_3_9__424_1 | Nouvelles annales de mathématiques, 1890, série 3, tome 9, p. 424 (NAM_1890_3_9__424_1) | 1,890 | und | fr | journal article | Numdam | http://www.numdam.org/item/NAM_1890_3_9__424_1/ | 3 | null | repository plate image | [
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hmb_01a8298eb2219198 | test | true | sg_96e6c7284e | bk_01a4a08037e8 | wk_betti_brioschi_problemi_e_teoremi_1880 | Betti Brioschi - Problemi e teoremi, 1880 | 1,880 | und | it | book | Wikimedia Commons | https://commons.wikimedia.org/wiki/File:Betti_Brioschi_-_Problemi_e_teoremi%2C_1880.pdf | 26 | null | page-scan crop | [
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Sia (Fig. I, II) AEF la retta che passa per A. Per C conducasi CG | EF, per D e B le DF, BE perpendiculari ad EF e si prolonghi BE ad incontrare CG in un punto G. Da questa costruzione risultano i triangoli ADF, CBG che hanno l'angolo F=G, DAF=BCG, il lato AD=BC, dunque sono uguali, e quindi DF=BG, onde BE+DF=... | 4590e364522d732a77aacd058157357c95a00788c4838e8c58f67b61c385e154 | us_public_domain_by_date | published 1880 (US public domain by date); Wikimedia Commons; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_80ef68fe18b8c6c3 | 44 | 17 | 20 | 7 | 90 | {
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hmb_01c0a2dd0a1245a7 | test | true | sg_c3804aedff | bk_f2a5f0f888ba | wk_smith_introductory_modern_geometry_point_ray_circle | Smith - Introductory Modern Geometry of Point, Ray, and Circle | 1,902 | und | en | book | Archival (author-held scan) | not recorded (owner waiver) | 196 | null | repository plate image | [
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hmb_01d368448eae4d52 | test | true | sg_c3804aedff | bk_555b85e5ee0c | wk_halsted_elements_of_geometry | Halsted - The Elements of Geometry | 1,885 | und | en | book | Archival (author-held scan) | not recorded (owner waiver) | 152 | null | repository plate image | [
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hmb_01f5baa3437b2635 | test | true | sg_ff2b0d52fa | bk_2eac8ea2e089 | wk_plane_and_solid_analytic_geometry | Plane and solid analytic geometry | 1,920 | und | en | book | Internet Archive | https://archive.org/details/planesolidanaly00osgo | 87 | null | page-scan crop | [
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ANALYTIC GEOMETRY
Let the triangle be $OAP$, with $M$ as the mid-point of $AP$. Let $a$ be the length of the base $OA$, and let $l$ be the given distance. It is required to find the locus of $P$, so that always
$$ \begin{array}{l}(1)\quad O M=l.\end{array} $$
It is convenient to take the origin of coordinates in O a... | afb76f466b022e4df4670de9d187b57e08708f473d0f3b48e308a79ec78ff8c4 | us_public_domain_by_date | published 1920 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_2839f41d618c6539 | 39 | 14 | 16 | 9 | 54 | {
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hmb_02595fd895e23bf1 | test | true | sg_75d88a25de | bk_0e566d9a1ce3 | wk_elements_of_analytic_geometry | Elements of analytic geometry | 1,897 | und | en | book | Internet Archive | https://archive.org/details/elementsofanalyt00hard | 355 | null | page-scan crop | [
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Surfaces
PROPOSITION I
433. A surface may be represented by an equation of the form
$$ f(x,y,z)=o. $$
Fig. 157
In Fig. 157 let MN represent any surface, and let OX, OY and OZ be the axes of coordinates.
From any point P on the surface draw PK $ \perp $ to the XY plane. It will generally cut the surface in ... | 1d472971a17b3105f9a605823719f5ea46e793cb725c95db11d960359c74a467 | us_public_domain_by_date | published 1897 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_30be4fca7ab8726d | 71 | 26 | 27 | 18 | 106 | {
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hmb_027affb5e3c70b27 | test | true | sg_38e410de05 | bk_1997df1c04fa | wk_elements_of_analytical_geometry_loomis | The elements of analytical geometry | 1,900 | und | en | book | Internet Archive | https://archive.org/details/elementsofanalyt00loom | 102 | null | page-scan crop | [
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In the equation $y^{2}=4a^{\prime}x$, $4a^{\prime}$ is the parameter of the diameter passing through the origin.
The parameter to the axis is called the principal parameter, or latu... | bfb22ea266b8c4725ff699ba7f3ba3b0c1dbc178284b3839b2b343130185acf8 | us_public_domain_by_date | published 1900 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_02dcd3d2cf27c74b | 51 | 20 | 23 | 8 | 75 | {
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आ | 28595dec7fec7525114d16cdbda1f3beeae2e841dd44a43350ee3ae0c06f02eb | us_public_domain_by_date | published 1826 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_bcea6bf392b371b3 | 23 | 11 | 12 | 0 | 28 | {
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118
Hence the tangent at the extremity of a diameter is parallel to its system of chords.*
This fact, combined with the relation between conjugate diameters, defined in Art. 83, enables us to readily draw any pair of conjugate diameters. Thus: at the extremity of any diameter draw a tangent to the ... | d68641fc1562d5527b5b9dc270316740a5599c216ce35fce40b46f92d6a98169 | us_public_domain_by_date | published 1907 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_56baca1e7ad74831 | 77 | 26 | 30 | 21 | 114 | {
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84. 四角壔之四對角線，過同一之點，則此四角壩爲平行六面體。
關 ABCD-EFGH為四角壩，其對角線 AF, BG, CH, DE,
過一點 O，則底 ABCD
與 EFGH 平行，且 AH
與 CF 平行，故 AH=
亦平行于其平行二直線，故EE'及FF',皆平行於XY，以下如前云云.
86. 自四面體之各角頂，連結相對面之重心之直線，相交於同一之點。
圈 E，F 为四面体 S-ABC 平面 ABC，SAB 之重心，连结 (E, SF, SE, CF，则 SE, CF，皆相交于 AB 之中点 D
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2°, если дуги не равны, то бобьшей изъ нихъ соотвътствуетъ ббльшій центральный уголь.
Доказательство отъ противнаго (илн наложеніем) предо-ставляемъ самимъ учащимся.
164. Теорема. Въ одномъ кругъ или въ равных кругахъ центральные углы относятся, как... | 02618e9022f4699591c6099a1386292223188ea274dada41901c7562db770899 | us_public_domain_by_date | published 1914 (US public domain by date); Wikimedia Commons; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_8cf01511ec751e5c | 53 | 22 | 26 | 5 | 62 | {
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hmb_02f76e474c93cb52 | test | true | sg_75d88a25de | bk_0e566d9a1ce3 | wk_elements_of_analytic_geometry | Elements of analytic geometry | 1,897 | und | en | book | Internet Archive | https://archive.org/details/elementsofanalyt00hard | 166 | null | page-scan crop | [
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I45
202. The Conjugate Axis.—A straight line drawn through the center perpendicular to the transverse axis, bisected by it and equal to twice the square root of the difference between the square of the distance from the focus to the center and the square of the semi-transverse axis is called the conjugate... | a626a45a70b4acb1840b560e434e191e049cc6db1b29a028862880e3a40bea66 | us_public_domain_by_date | published 1897 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_3df90df66948550e | 37 | 16 | 14 | 7 | 69 | {
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hmb_02f8018d84e45605 | test | true | sg_2770a91e4b | bk_0de514b2f707 | wk_recorde_pathway_to_knowledge | Recorde - The Pathway to Knowledge | 1,551 | und | en | book | Project Gutenberg | https://www.gutenberg.org/ebooks/33093 | 107 | null | transcriber-prepared image | [
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hmb_030f49f7c0dda1d6 | test | true | sg_c3804aedff | bk_555b85e5ee0c | wk_halsted_elements_of_geometry | Halsted - The Elements of Geometry | 1,885 | und | en | book | Archival (author-held scan) | not recorded (owner waiver) | 392 | null | repository plate image | [
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hmb_035591b93e268af3 | test | true | sg_75d88a25de | bk_eb2821092a1e | wk_first_course_in_analytical_geometry_schmall | A first course in analytical geometry | 1,921 | und | en | book | Internet Archive | https://archive.org/details/firstcourseinana00schm | 53 | null | page-scan crop | [
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ANALYTICAL GEOMETRY
Ex. 19. Plot the curve $ y = \tan x $. | f604f3e6720bef1ec273c248ae929563c84fea35d315bb92918e384cf2191fde | us_public_domain_by_date | published 1921 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_d069781908d118d0 | 33 | 15 | 13 | 5 | 50 | {
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hmb_035da63b77eda3f0 | test | true | sg_165e3c8a01 | bk_38a4f0c22bf8 | wk_descartes_geometrie | Descartes - La géométrie | 1,886 | und | fr | book | Project Gutenberg | https://www.gutenberg.org/ebooks/26400 | 15 | null | transcriber-prepared image | [
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hmb_037ff9144069d007 | test | true | sg_ff2b0d52fa | bk_2eac8ea2e089 | wk_plane_and_solid_analytic_geometry | Plane and solid analytic geometry | 1,920 | und | en | book | Internet Archive | https://archive.org/details/planesolidanaly00osgo | 63 | null | page-scan crop | [
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1. Taking the same triangle as before, choose the axis of x along the side AB, but take the axis of y through A. The coordinates of the vertices will then be:
$$ \mathcal{A}:(0,0);\quad B:(6,0);\quad C:(2,6). $$
Prove the theorem for this triangle.
2. The vertices of a triangle lie at the points $(0, 0)$, $(3... | 83ee1a7f937ab75f815cbcc99dad744637b6f7bdeea61ccdc91b2cf2fed69ed1 | us_public_domain_by_date | published 1920 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_62ae4541c276a67f | 57 | 17 | 23 | 17 | 93 | {
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hmb_039993b1006d4d6b | test | true | sg_b20be833cc | bk_fb78b95482ba | wk_on_teaching_geometry | On teaching geometry | 1,900 | und | en | book | Internet Archive | https://archive.org/details/onteachinggeomet00milnrich | 27 | null | page-scan crop | [
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Proof. 1. Draw the auxiliary line $ FG $, and let HK be drawn from $ H \perp $ to AB.
2. At a given point in a straight line a perpendicular to the line can be drawn, and but one. (25)
$ \therefore $ 3. HK is $ \perp $ to FG at H.
Apply the ... | 39bbd03b19e0d72f9856384870825463df676efa724dacfe2286c15a45d3dab8 | us_public_domain_by_date | published 1900 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_ef79b3e955bcc8d4 | 49 | 16 | 26 | 7 | 56 | {
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hmb_039b606572b3b2af | test | true | sg_4b55a27539 | bk_aeee123c9d38 | wk_cantor_vorlesungen_geschichte_mathematik_bd2 | Cantor - Vorlesungen über Geschichte der Mathematik, Bd. 2 | 1,892 | de | de | book | GDZ Göttingen | https://gdz.sub.uni-goettingen.de/id/PPN517628244 | 183 | null | repository plate image | [
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hmb_03a60b1b0c391049 | test | true | sg_75d88a25de | bk_eb2821092a1e | wk_first_course_in_analytical_geometry_schmall | A first course in analytical geometry | 1,921 | und | en | book | Internet Archive | https://archive.org/details/firstcourseinana00schm | 165 | null | page-scan crop | [
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Fig. 110.
MN = 2a and draw PT ⊥ to PN; or take OT = OM and draw PT. In either way PT will be tangent at P. [Why?]
Fig. 111.
(2) To draw a tangent to a parabola from an external point P. With P as center, and PF as radius, cut the directrix in R. Draw RQ ... | f8f59bdb2d8d7ddd28e40c0a80d16c8135c808704dba927823e7adfdd270e64c | us_public_domain_by_date | published 1921 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_d9bdd7079a3fa08c | 45 | 20 | 19 | 6 | 72 | {
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hmb_03cebc3fd98b94a0 | test | true | sg_2cb2026b8c | bk_19b7102a59e5 | wk_efremov_d_novaia_geometriia_treugolnika_1902 | Ефремов Д. Новая геометрия треугольника (1902) | 1,902 | und | ru | book | Wikimedia Commons | https://commons.wikimedia.org/wiki/File:%D0%95%D1%84%D1%80%D0%B5%D0%BC%D0%BE%D0%B2_%D0%94._%D0%9D%D0%BE%D0%B2%D0%B0%D1%8F_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F_%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA%D0%B0_%281902%29.pdf | 13 | null | page-scan crop | [
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$$ \frac{\mathrm{A^{\prime}D}}{\mathrm{A^{\prime}K}}=\frac{\mathrm{BG}}{\mathrm{BA}}=\frac{\mathrm{A^{\prime}K}}{\mathrm{A^{\prime}B^{\prime}}}, $$
TO
$$ \mathrm{A^{\prime}D.A^{\prime}B^{\prime}=A^{\prime}K^{2}=A^{\prime}\alpha^{2... | 60cb01ec39f67ed96b2d9803c8ec3861e7bfb7ed4b2ffcf35c773f627e674b0d | us_public_domain_by_date | published 1902 (US public domain by date); Wikimedia Commons; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_9dfd149d596f281e | 83 | 35 | 31 | 17 | 155 | {
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Se in un cerchio siano inscritti il pentagono, l'esagono ed il decagono equilateri, il quadrato del lato del pentagono è uguale alla somma dei quadrati dei lati dell'esagono e del decagono. (Fig. 163 e 164).
Fig. 164.
Proponiamoci prima di risolvere il seguente problema. Calcolare in funzione del razgio il lato de... | b5e3f9f0a490e47aa8221177439a7e1ce808ad8f9e7c15263f183d5f9db8688e | us_public_domain_by_date | published 1880 (US public domain by date); Wikimedia Commons; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_fe2da028806c62eb | 48 | 17 | 25 | 6 | 64 | {
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127
$ ^{16}y^2 + 9x^2 = 144 $. Find the slope of the chord which joins the two points of tangency. Ans. Slope = $ -\frac{9}{8} $.
2. Two tangents are drawn to $16y^{2}+9x^{2}=144$ at the extremities of the chord $y=-\frac{3}{4}x+3$. Where do the tangents meet?
Ans. $x=4, y=3$.
Construct a figure showing th... | 3feca5a93f6368eedafb7a7db105fcf7c14f4ef04cecfc9f370232254bb6aa26 | us_public_domain_by_date | published 1897 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_c518d621258fe5f8 | 65 | 27 | 24 | 14 | 113 | {
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THE PARABOLA.
$$ y y^{\prime}=2ax-2ax^{\prime}+4ax^{\prime}, $$
or
$$ y y^{\prime}=2a(x+x^{\prime}), $$
which is the simplest form of the equation to the tangent line.
92. Points where the tangent cuts the axes. To determine
the point in which the tangent intersects the axis of X, we make y=0, which gives
$$ 0=2a(x+... | f0fa61478e87426fb9644bd61526b254f7ff1c9f63c603dbcc7257cc0e33c7dc | us_public_domain_by_date | published 1900 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_cb1086cc5f4f3219 | 26 | 10 | 11 | 5 | 42 | {
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hmb_04a382c39fd155ab | test | true | sg_4b55a27539 | bk_aeee123c9d38 | wk_cantor_vorlesungen_geschichte_mathematik_bd2 | Cantor - Vorlesungen über Geschichte der Mathematik, Bd. 2 | 1,892 | de | de | book | GDZ Göttingen | https://gdz.sub.uni-goettingen.de/id/PPN517628244 | 120 | null | repository plate image | [
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hmb_04aa0e3a7985c3d1 | test | true | sg_75d88a25de | bk_eb2821092a1e | wk_first_course_in_analytical_geometry_schmall | A first course in analytical geometry | 1,921 | und | en | book | Internet Archive | https://archive.org/details/firstcourseinana00schm | 150 | null | page-scan crop | [
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133
$$ \sqrt{(x-1)^{2}+(y-1)^{2}}=\frac{3}{5}\left(\frac{2x-y-1}{\pm\sqrt{5}}\right) $$
Here
is the required equation.
Ex. 3. Find the equation of the hyperbola focus $ (2,0) $, directrix x-9=0, eccentricity $ \frac{3}{2} $.
$$ \sqrt{(\dot{x}-2)^{2}+y^{2}}=\frac{3}{2}\left(x\div9\right)\text{is the requi... | 10f33f5d542b3fd2cef56e85bd2d6ea4985df200b0bfa40d5dd777fdbd446179 | us_public_domain_by_date | published 1921 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_2d0f919084c6bea9 | 61 | 20 | 29 | 12 | 77 | {
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hmb_04ab2e328ecb901a | test | true | sg_75d88a25de | bk_0e566d9a1ce3 | wk_elements_of_analytic_geometry | Elements of analytic geometry | 1,897 | und | en | book | Internet Archive | https://archive.org/details/elementsofanalyt00hard | 298 | null | page-scan crop | [
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277
Fig. 129
FOURTH CASE
$$ \mathrm{Let}s=2d. $$
$ -\frac{s^{2}}{4d} = -d \text{ and } d - s = -d. $
Then
Now when $x = -d$ in [8], $b = 0$ and $c = 0$ and the four values of $y$ in [6] become 0.
From—d to 3d, y hastwo real and equal values with opposite signs. Therefore the limaçon takes the form of Fig. 13... | 61164bfd5b0f367e555b40c91bc2e0d154a1ccadcf22f0cdace0528773cfc360 | us_public_domain_by_date | published 1897 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_81928485f3b1a360 | 12 | 5 | 4 | 3 | 23 | {
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hmb_04e24021f45b6966 | test | true | sg_c3804aedff | bk_75b06a64f496 | wk_shuxue_cidian | SSID-11317445 數學辭典 | 1,933 | und | zh | book | Wikimedia Commons | https://commons.wikimedia.org/wiki/File:SSID-11317445_%E6%95%B8%E5%AD%B8%E8%BE%AD%E5%85%B8.pdf | 529 | null | page-scan crop | [
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CF=BD: CG, AE:
EC = AD : CG. 但
AE: EC = BF : CF,
故 BD : CG = AD :
CG，因而 BD = AD.
自 CB 之中點 M 及 C，引 EDF 之平行線 $ E^{\prime}MF^{\prime} $ 及 CG，則
$ CE'=GF'=BF' $. 而
$ \triangle BDF \sim \triangle AMF' $, $ \triangle CED $
$ \triangle CE'M $，故 $ \frac{BD}{BF}= $
$ \frac{BM}{BF'},\quad\frac{CD}{CE}=\frac... | ca396b610c3aad281197f1c603a81c68d7a6d2d4bb56ba182e646c8747d773ba | owner_determination | published 1933; US public-domain status recorded by the project owner; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_8f6bf3a253d687f7 | 49 | 19 | 24 | 6 | 89 | {
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hmb_04e76a2e04a46437 | test | true | sg_318e34f1a7 | bk_ed1f0c34aac3 | wk_analytic_geometry_and_principles_of_algebra | Analytic geometry and principles of algebra | 1,913 | und | en | book | Internet Archive | https://archive.org/details/analyticgeometry00ziwerich | 145 | null | page-scan crop | [
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Fig. 52
(1) If $a>0$, the curve lies above the axis $Ox$. For, no matter what positive or negative value is assigned to $x$, $y$ is positive. Furthermore, as $x$ is allowed to increase in abso... | cda8e551ae91c04ce543a3100b1cdaec2c078a8ec0c350b2cfe3141c2d4abcf0 | us_public_domain_by_date | published 1913 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_04f646bd17ddda7c | 11 | 4 | 4 | 3 | 11 | {
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hmb_05229043dbf3f839 | test | true | sg_75d88a25de | bk_0e566d9a1ce3 | wk_elements_of_analytic_geometry | Elements of analytic geometry | 1,897 | und | en | book | Internet Archive | https://archive.org/details/elementsofanalyt00hard | 62 | null | page-scan crop | [
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41
Fig. 21
Let MN be any straight line. Let OR be a straight line drawn from the origin perpendicular to MN.
$$ \mathrm{L e t}\not{p}\equiv\mathrm{O R~a n d~}\alpha\equiv\angle\mathrm{R O A}. $$
Let P be any point on the line MN and draw its coordinates PK and OK.
$$ Let x\equiv OK and y\equiv PK. $$... | 5c101421bdd8938fe98693c8d300c3c0cf5cfbc8f2d8187f01dae7f4f1961a45 | us_public_domain_by_date | published 1897 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_4966ac2d0eb43d76 | 77 | 26 | 36 | 15 | 107 | {
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hmb_05463ec15d86ffb7 | test | true | sg_5ca4ca2421 | bk_38336e23530c | wk_nam_1900_3_19__255_1 | Nouvelles annales de mathématiques, 1900, série 3, tome 19, p. 255 (NAM_1900_3_19__255_1) | 1,900 | und | fr | journal article | Numdam | http://www.numdam.org/item/NAM_1900_3_19__255_1/ | 1 | null | repository plate image | [
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hmb_0559970452cad21d | test | true | sg_b4f36e9a13 | bk_5783f0c55359 | wk_cantor_vorlesungen_geschichte_mathematik_bd1 | Cantor - Vorlesungen über Geschichte der Mathematik, Bd. 1, 2. Aufl. | 1,894 | de | de | book | GDZ Göttingen | https://gdz.sub.uni-goettingen.de/id/PPN51762818X | 2 | null | repository plate image | [
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] | imgs/p00069/img_in_image_box_51_184_319_339.jpg | 54b5c159c09852e12156ab330a5edab3ec0b406f668f5633fc1edb1c46e387ac | geometry | predicted by global.openai.gpt-6-luna | false | singleton | [] | c5f8ed9de94e73955fee076cba86d2d24fdc5492c4c0ad25f8af31c15cc7ff1e | 1,024 | 1,024 | 2b7e26823cdc7c79f6986dd387e7da87fe5fa83016e0e8e5acda8ebd0a8a5289 | false | 0 | null | e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855 | us_public_domain_by_date | published 1894 (US public domain by date); GDZ Göttingen; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_9e806b6b8786fdc2 | 71 | 23 | 40 | 8 | 89 | {
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hmb_05702f0159bdc455 | test | true | sg_ceda458e67 | bk_60c7bf37015b | wk_plane_and_solid_analytic_geometry_an_elementary_textbook | Plane and solid analytic geometry; an elementary textbook | 1,901 | und | en | book | Internet Archive | https://archive.org/details/planesolidanaly00asht | 46 | null | page-scan crop | [
232,
374,
501,
570
] | paddle_preprocessed | [
536,
822
] | [
"dewarped",
"resampled",
"jpeg_recompressed",
"paddle_bounds"
] | imgs/p00047/img_in_image_box_232_374_501_570.jpg | 6e219f80e410d002f850cd052bf8341111e0a044583776d950ed9890f8262f25 | graph_plot | predicted by global.openai.gpt-6-luna | true | singleton | [] | cabaf8248ab2e8bf79e68efd78735e66e7eb8df08db16d42a9c9958102a5927a | 1,024 | 1,024 | f29ec771026d809545f5afd5ded21e0a1c3b1d8d43bde001f32b18152ece7dc0 | true | 1,072 | $$ x^{2}-y^{2}=25. $$
Solving the equation for $y$, we have $y=\pm\sqrt{x^{2}-25}$, from which it appears that $y$ is imaginary, so long as $-5<x<+5$. There will therefore be no points on the locus for which $x$ is numerically less than 5.
$$ \begin{aligned}If\quad&x=5,y=0；&x=-5,y=0；\\&x=6,y=\pm\sqrt{11}；&x=-6,y=\pm\sq... | 46f4b6397ed7286c60f4549de5e4dcb2444b78abb92ddf44bbd10b346bb2a685 | us_public_domain_by_date | published 1901 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_a8c1688b8748247f | 36 | 18 | 13 | 5 | 42 | {
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hmb_05a9cdb3a03ec46e | test | true | sg_2cb2026b8c | bk_19b7102a59e5 | wk_efremov_d_novaia_geometriia_treugolnika_1902 | Ефремов Д. Новая геометрия треугольника (1902) | 1,902 | und | ru | book | Wikimedia Commons | https://commons.wikimedia.org/wiki/File:%D0%95%D1%84%D1%80%D0%B5%D0%BC%D0%BE%D0%B2_%D0%94._%D0%9D%D0%BE%D0%B2%D0%B0%D1%8F_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F_%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA%D0%B0_%281902%29.pdf | 61 | null | page-scan crop | [
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616
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739,
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] | [
"dewarped",
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"jpeg_recompressed",
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] | imgs/p00062/img_in_image_box_5_281_373_616.jpg | 458e05c708e730a79c3002a2c331fa26c5b95406e9f0c2ff7de11ad0df06bcef | geometry | predicted by global.openai.gpt-6-luna | false | singleton | [] | 1482e2b9222ea876d893d0437ee184515e343e2d5b31194b8fa8ae093186f659 | 1,024 | 1,024 | d6ec399cdf1b36b0156d68a23002cc22045538c4f70b733c8b37860e285ab72e | true | 1,101 | слідовательно
$$ (\mathbf{Z},\mathbf{Y A C^{\prime}A^{\prime}})=-(\mathbf{Z},\mathbf{X A C^{\prime}A^{\prime}}); $$
а потому Zy и ZX составляють одну прямую, т. е. тр-ки ABC и АВС-гомологичны.
17. Теорема. Паппа (Pappus). Произведение разстояній какой-либо точки окружности отв двух противоположных стороны вписанного че... | b14898e33db18303b3af5db4b979f07362d14cb25cfc02bf555cd7eb6b97b60d | us_public_domain_by_date | published 1902 (US public domain by date); Wikimedia Commons; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_8c4a6d520576b109 | 78 | 37 | 27 | 14 | 125 | {
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hmb_05f113cd7fce5b96 | test | true | sg_883a4bb4fa | bk_8add960606fc | wk_clairaut_elements_de_geometrie_1771 | Clairaut - Elementos de geometria | 1,772 | und | pt | book | Project Gutenberg | https://www.gutenberg.org/ebooks/78897 | 11 | null | transcriber-prepared image | [
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262,
107
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"cropped_by_collector",
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hmb_062e1842f637f42f | test | true | sg_42aaee2473 | bk_b9f5ae14d11b | wk_goojratee_version_of_a_course_of_practical_geometry | Goojratee Version of a Course of Practical Geometry | 1,826 | und | gu | book | Internet Archive | https://archive.org/details/dli.granth.35738 | 82 | null | page-scan crop | [
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"dewarped",
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hmb_067e3b8654ee8751 | test | true | sg_f0f9285215 | bk_7cde0e16545e | wk_ebene_geometrie_der_lage | Ebene geometrie der lage | 1,900 | und | de | book | Internet Archive | https://archive.org/details/ebenederlagevon00bogerich | 218 | null | page-scan crop | [
252,
222,
584,
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634,
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] | [
"dewarped",
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] | imgs/p00219/img_in_image_box_252_222_584_549.jpg | 3dd55592b8762af807fb316eba32d7f5a216f3765ecc6b7a73962ae6acc66a70 | geometry | predicted by global.openai.gpt-6-luna | true | singleton | [] | 5bf613d85aa27aaca12ca239d941fbf5f2fa17b1d6c9aa652afa62f31193983f | 1,024 | 1,024 | b2278d8f45b8b4c22a6a94cad687be597f6289837bd17d4d9d03ccaf6a99a98e | true | 1,051 | § 14. Konjugierte Projektivitäten. Nr. 175—176.
203
Tangente in S von PA geschnitten wird, und schneidet die Kurve in einem Punkte B, der der Berührungspunkt der zweiten von E an die Kurve gezogenen Tangente $ \beta_{1} $ ist $ ^{(86}\mathrm{Z}_{1)} $; die Verbindungslinie AB geht durch $ P^{(98)} $. — Ist nun $ B_{... | 7bdfadf4d2a9c19f217388976b1ee922ecc91919619f0a1a9940b67b263ff387 | us_public_domain_by_date | published 1900 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_01b511ac4f12ef7b | 54 | 24 | 15 | 15 | 91 | {
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hmb_068123e9a105996d | test | true | sg_96e6c7284e | bk_01a4a08037e8 | wk_betti_brioschi_problemi_e_teoremi_1880 | Betti Brioschi - Problemi e teoremi, 1880 | 1,880 | und | it | book | Wikimedia Commons | https://commons.wikimedia.org/wiki/File:Betti_Brioschi_-_Problemi_e_teoremi%2C_1880.pdf | 196 | null | page-scan crop | [
36,
613,
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714,
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"dewarped",
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] | imgs/p00197/img_in_image_box_36_613_171_751.jpg | 80fa12a84f8e7dd26aa8e088a6ae3adeae4ee4f79ac02207c3652ca6252e9253 | geometry | predicted by global.openai.gpt-6-luna | false | singleton | [] | 64be033bd6e519deaabe511d9ed6a5d46454211ccf0600df031d00e06b63c056 | 1,024 | 1,024 | cbda996f1a35e34ef6d98e09ba3fa3ef1c42fc4d22fefb63fa7eacfbaf2767e7 | true | 797 | 13.
I piani che distano ugualmente dal centro della sfera danno luogo a sezioni uguali. Fra due sezioni disuguali la maggiore è quella prodotta dal piano più prossimo al centro. (Fig. 183).
Fig. 183.
1.° Siano AB, CD i due piani (che sono due cerchi (Es. prec.)) ugualmente lontani dal centro O della sfera. E chiaro che... | 1e9e1d0eabfc0d2185dfd5789ea782dd36719e02a330df6302127bdc86f950de | us_public_domain_by_date | published 1880 (US public domain by date); Wikimedia Commons; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_2232992744a88d12 | 56 | 22 | 25 | 9 | 89 | {
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hmb_0694d980ffc6be22 | test | true | sg_eb8da40a87 | bk_6d93714e3ebd | wk_nam_1908_4_8__390_0 | Nouvelles annales de mathématiques, 1908, série 4, tome 8, p. 390 (NAM_1908_4_8__390_0) | 1,908 | und | fr | journal article | Numdam | http://www.numdam.org/item/NAM_1908_4_8__390_0/ | 6 | null | repository plate image | [
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] | [
"cropped_by_collector",
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] | imgs/p00010/img_in_image_box_281_345_437_463.jpg | 2825c706111226742f060f05e864a7443b2fe0196487f3d88cf668ab9b5ce137 | geometry | predicted by global.openai.gpt-6-luna | true | singleton | [] | a7a973056179df54f08618ef6dc1efdec8cabf5f988b4d680979ed7eaebf0cd7 | 1,024 | 1,024 | c6654b7f59f6d1619858a84847d3463586f355dc9707f41a00ce4ccb8ca5f321 | false | 0 | null | e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855 | us_public_domain_by_date | published 1908 (US public domain by date); Numdam; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_9c6e1bf2abb7409a | 15 | 5 | 7 | 3 | 17 | {
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hmb_0696784ce7e23a72 | test | true | sg_4b55a27539 | bk_aeee123c9d38 | wk_cantor_vorlesungen_geschichte_mathematik_bd2 | Cantor - Vorlesungen über Geschichte der Mathematik, Bd. 2 | 1,892 | de | de | book | GDZ Göttingen | https://gdz.sub.uni-goettingen.de/id/PPN517628244 | 154 | null | repository plate image | [
0,
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319,
235
] | precropped_image | [
319,
235
] | [
"cropped_by_collector",
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"jpeg_recompressed",
"paddle_bounds"
] | imgs/p00650/img_in_image_box_392_454_711_689.jpg | 387c1cf46acaf43190a4206343e7dcf982e818ad0a8eb2e8db3880603150c078 | geometry | predicted by global.openai.gpt-6-luna | false | singleton | [] | 4609d40d76cec5a55023635cc10ae33af8a4629d67546660b5eae6a9de730caf | 1,024 | 1,024 | b81d50e6b3ff152e30237b64ff0197f51c887ac6861d3c90722edc8cc56182c4 | false | 0 | null | e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855 | us_public_domain_by_date | published 1892 (US public domain by date); GDZ Göttingen; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_06a229ddce7fa151 | 52 | 18 | 23 | 11 | 69 | {
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hmb_069d9989b1a687da | test | true | sg_c3804aedff | bk_f2a5f0f888ba | wk_smith_introductory_modern_geometry_point_ray_circle | Smith - Introductory Modern Geometry of Point, Ray, and Circle | 1,902 | und | en | book | Archival (author-held scan) | not recorded (owner waiver) | 64 | null | repository plate image | [
0,
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361,
231
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361,
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] | [
"cropped_by_collector",
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] | imgs/p00063/img_in_image_box_426_486_787_717.jpg | d94f24459591313d04305a53333aa5620771f7354d636f855528057fe1725fb9 | geometry | predicted by global.openai.gpt-6-luna | true | singleton | [] | 5b092dfb44efa602f82e15ce4879ce04f3b9848539b6f1cf401820d45e340d14 | 1,024 | 1,024 | d292633847d246aab92e72059b17d060ce7ba6eb1ef6e2ae4a845da6793ea492 | false | 0 | null | e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855 | author_held_scan | cleaned scan held by the authors; published 1902; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_2ca312767fc51e95 | 34 | 14 | 16 | 4 | 39 | {
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hmb_06be7d9982f20ad2 | test | true | sg_c3804aedff | bk_f2a5f0f888ba | wk_smith_introductory_modern_geometry_point_ray_circle | Smith - Introductory Modern Geometry of Point, Ray, and Circle | 1,902 | und | en | book | Archival (author-held scan) | not recorded (owner waiver) | 78 | null | repository plate image | [
0,
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197,
404
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197,
404
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149
Let P'AL be the equilateral hyperbola having the same transverse axis.
Draw the ordinates P'K and PK to the same point K.
$$ Let x\equiv CK,y\equiv PK and y^{\prime}\equiv P^{\prime}K. $$
$$ \begin{array}{l}Let a\equiv\mathrm{CA}and b\equiv\mathrm{CB}.\end{array} $$
We are to prove that
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di cerchio si tiri una perpendicularare al diametro che passa per una delle sue estremità, questa perpendicularare bisecherà il segmento fatto nella corda dalla retta che congiunge il medesimo punto dell'arco coll'altra estremità del medesimo diametro. (Fig. 74).
Sia DC l'arco bisecato in A, AFE la perpendiculare al... | 2f802a6c69602231cb7229011a3ec9ec529613be2b42699bcac4bb8c477154f3 | us_public_domain_by_date | published 1880 (US public domain by date); Wikimedia Commons; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_dbfc9d841f599c23 | 46 | 18 | 20 | 8 | 78 | {
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TRANSCENDENTAL CURVES.
Thus, let PD be a straight line which revolves uniformly
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221
inclined to the base. Let APB be such a section, and let ABC be a section of the cylinder through its axis, and perpendicular to the plane of the former section. Draw a plane perpendicular to the axis of the cylinder, intersecting the cylinder in a circle whose diameter is DE, and interse... | f09a5062e3c91f58c282a1d7faf19f2de19aae036d55174598d9145ae60d67e9 | us_public_domain_by_date | published 1900 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_70082ad3e1d3cbf3 | 54 | 22 | 25 | 7 | 86 | {
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THE TEACHING OF GEOMETRY
Make $BQ$ and $AQ'$ perpendicular to $AB$, and make $BP = PQ = AP' = P'Q$. Then $\triangle XYZ$ is congruent to $\triangle YBP$, and also to $\triangle XAP$. Therefore $AX = XY = YB$. In the same way we might continue to produce $BQ$ until it is made up of $n - 1$ lengths $BP$, and so for $... | f54f68fac8ef80cf3bde4517aea876705c451a9ca37e30a73247fd932e3a3c22 | us_public_domain_by_date | published 1911 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_acce5fe1e6d45578 | 60 | 25 | 24 | 11 | 97 | {
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hmb_0790b826413a125f | test | true | sg_165e3c8a01 | bk_38a4f0c22bf8 | wk_descartes_geometrie | Descartes - La géométrie | 1,886 | und | fr | book | Project Gutenberg | https://www.gutenberg.org/ebooks/26400 | 14 | null | transcriber-prepared image | [
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DYNAMIC SYMMETRY
Fig. 31.
Fig. 31a.
Later, in XIII, 18, the rectangular relationship is more clearly shown in a root-five rectangle. The Euclidean diagram of the 18th proposition is peculiarly interesting in the light of dynamic symmetry because it suggests what may have been the Greek method of constructing the dyn... | f96542b5649c895d340498fa31a473ad2f1d1fb8b96e17a2710d5b9996e908e8 | us_public_domain_by_date | published 1920 (US public domain by date); Wikimedia Commons; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_28cc71d704927dd9 | 19 | 9 | 10 | 0 | 29 | {
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hmb_07e00fe21dad2bd4 | test | true | sg_ceda458e67 | bk_60c7bf37015b | wk_plane_and_solid_analytic_geometry_an_elementary_textbook | Plane and solid analytic geometry; an elementary textbook | 1,901 | und | en | book | Internet Archive | https://archive.org/details/planesolidanaly00asht | 200 | null | page-scan crop | [
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181
are perpendicular to each other, k = -2. Substituting this value in the equation and transposing, it becomes
$$ (x-y-2)^{2}=4\left(x+y-3\right). $$
Dividing both members by $ a^2 + c^2 $, and both dividing and multiplying the second member by $ \sqrt{D^{n_2} + E^{n_2}}... | 3571f927201aab954fae0424d49833c41fb396fd5d397b637a3a9cde63b3fc39 | us_public_domain_by_date | published 1901 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_da5adeac59eda643 | 27 | 10 | 10 | 7 | 30 | {
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The graph of the function x is the infinite straight line passing through the origin and bisecting the angles of the first and the third quadrant.
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] | imgs/p00189/img_in_image_box_292_633_457_797.jpg | e7787e17b49ef12a80006701ca82252584d085a7f5150f133e192c125daba232 | geometry | predicted by global.openai.gpt-6-luna | true | singleton | [] | 1c87ab37c3fafd32615f65c6b886a5bf2ab196e3a46432c489fccfca6a2aa9dd | 1,024 | 1,024 | 852f298feae6144724b3c2b8fc509ca8eab1601cfa36041898242b74b0743029 | true | 585 | point A as the origin of co-ordinates, and let us suppose that the generating point has described the arc AP. If N designates the point at which the generating circle touches the base, it is plain that the line AN will be equal to the arc PN. Through N draw the diameter EN, which will be perpendicular to the base of th... | 55fe27be0cae694bc605344777139cf3854c83369e7d4988d6e184431faa60ee | us_public_domain_by_date | published 1900 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_17a21354cb3889e0 | 45 | 15 | 24 | 6 | 57 | {
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hmb_092858635a9ae0be | test | true | sg_fa35de2f3d | bk_49f0cb2aaadb | wk_analytic_geometry_for_technical_schools_and_colleges | Analytic geometry, for technical schools and colleges | 1,897 | und | en | book | Internet Archive | https://archive.org/details/analyticgeometry00lambrich | 66 | null | page-scan crop | [
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51
time is $ t=\frac{2\pi}{\omega} $. The construction of the curve is indicated in the figure. The projection of uniform motion in the circumference of a circle on a diameter is called harmonic motion.
Fra. 41.
To add graphically two sine-functions of equal periods $ y_1 = a_1 \s... | b20fb2e005f6409a6bb269cd0baeeba3f7153607537a72ab9648f29c69fd5afa | us_public_domain_by_date | published 1897 (US public domain by date); Internet Archive; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_d5f58ecdcfe83409 | 13 | 6 | 4 | 3 | 22 | {
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hmb_095171c901f20857 | test | true | sg_96e6c7284e | bk_01a4a08037e8 | wk_betti_brioschi_problemi_e_teoremi_1880 | Betti Brioschi - Problemi e teoremi, 1880 | 1,880 | und | it | book | Wikimedia Commons | https://commons.wikimedia.org/wiki/File:Betti_Brioschi_-_Problemi_e_teoremi%2C_1880.pdf | 95 | null | page-scan crop | [
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Di tutti i triangoli che hanno la stessa base e lo stesso angolo al vertice, il triangolo isoscele è il massimo; e reciprocamente di tutti i triangoli sulla stessa base e tra le medesime parallele, l'isoscele ha il massimo angolo al vertice. (Fig. 84).
Fig. 84.
1.º Siano ABC, DBC due triangoli cogli angoli A e D ug... | 4c46d795346e3cdd4a504430cd8e70292dd100ea6e0bfa5e915af189ed90a072 | us_public_domain_by_date | published 1880 (US public domain by date); Wikimedia Commons; The project owner states that permission to redistribute the corpus has been granted (statement of 2026-10-17). | card_7b6927da55275a06 | 41 | 19 | 17 | 5 | 75 | {
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hmb_0983c4a02f1ed863 | test | true | sg_4b55a27539 | bk_aeee123c9d38 | wk_cantor_vorlesungen_geschichte_mathematik_bd2 | Cantor - Vorlesungen über Geschichte der Mathematik, Bd. 2 | 1,892 | de | de | book | GDZ Göttingen | https://gdz.sub.uni-goettingen.de/id/PPN517628244 | 50 | null | repository plate image | [
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